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Thrice

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- Thread starter Thrice
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Thrice

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- #2

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What is he specifically referring to?

The notions of vectors and dual-vectors?

The notions of vectors and dual-vectors?

- #3

Thrice

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- #4

coalquay404

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There's nothing particularly great about Dirac's notation outside of, say, quantum mechanics and linear algebra. For example, given some vector space [itex]V(n,\mathbb{F})[/itex] with dual [itex]V^*(n,\mathbb{F})[/itex], we could choose to use Dirac's notation to denote an inner product between [itex]|\psi\rangle\in V(n,\mathbb{F})[/itex] and [itex]\langle\phi|\in V^*(n,\mathbb{F})[/itex] by

[tex]\langle\phi|\psi\rangle\in\mathbb{F}[/tex]

However, in index notation you would choose a basis [itex]\{e_i\}[/itex] for [itex]V(n,\mathbb{F})[/itex] and a corresponding dual basis [itex]\{\omega^i\}[/itex] for [itex]V^*(n,\mathbb{F})[/itex]. Then the inner product of the above quantities is

[tex]\phi_i\psi^i\in\mathbb{F}[/itex]

Dirac's notation is fine for dealing with vectors and their duals. However, things quickly become cumbersome when you deal with tensor products. It's not uncommon for one to deal with tensors of rank four and above; in coordinate free notation this would simply be [itex]\mathbf{T}(W,X,Y,Z)[/itex], while in index notation it's just [itex]T^{ijkl}[/itex]. Contrast that with it's representation in Dirac notation:

[tex]\mathbf{T}\to

|W\rangle\otimes|X\rangle\otimes|Y\rangle\otimes|Z\rangle.[/tex]

See? It's too cumbersome to bother with. As with all of these kinds of things, notation is just a tool: you pick the one most suited to the job at hand.

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- #5

Thrice

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Now that's interesting. I remember doing tensors as operators, but I never thought of index notation in QM. I'm guessing we mostly use the dirac notation because of historical reasons?[tex]\phi_i\psi^i\in\mathbb{F}[/itex]

- #6

coalquay404

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Now that's interesting. I remember doing tensors as operators, but I never thought of index notation in QM. I'm guessing we mostly use the dirac notation because of historical reasons?

I think it's a mixture of historical reasons (particularly Dirac's early writings, which are, by any standards, monumental) and geographic location. Anybody studying at UT, Austin probably uses [itex](\phi,\psi)=(\psi,\phi)^*[/itex] to denote inner products: it seems to be Weinberg's notation of choice.

Also, it's important to note the context in which Dirac's notation is particularly useful. In quantum mechanics, states are represented by rays in a complex separable Hilbert space. The operative word here is

[tex]\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*[/itex]

where the star denotes complex conjugation. I'm not sure if (a) it's really necessary to use Dirac notation in, say, general relativity since you rarely need to deal with complexified GR and (b) if it's really practical to deal with conjugation acting on multiple indices using stars.

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- #7

quantum123

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The super and sub-scripts are untidy, so much so that Penrose have to invent a string diagram where he draw strings to join all the indices.

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